In terms of the fixed non-negative integers $\alpha$ and $\beta$ determine the least upper bound of the ratio (or show that it is unbounded) \[ \frac{S(n)}{S(2^{\alpha}5^{\beta}n)} \]as $n$ varies through the positive integers, where $S(\cdot)$ denotes sum of digits in decimal representation.
Problem
Source: Bulgaria EGMO TST 2019 Day 2 Problem 3
Tags: number theory, sum of digits, Upper Bound on Sequence, ratio